Download Problem set 6

MATH 210
Problem Set 6
Due: February 26, 2016
Exercises
1. (a) Find a (simultaneous) solution to the system of congruences
f (x) ≡ x3 + 7
2
f (x) ≡ x − 13
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(mod x − 1),
(mod x − 2),
f (x) ≡ x + x + 1
(mod x − 3).
in Q[x].
(b) To gain an appreciation for how nice univariate polynomial rings are, consider
f (x, y) = y 2 − x3 − x − 1 ∈ C[x, y].
Show that f (x, y) is irreducible. (Compared to the fundamental theorem of arithmetic, this is
quite a change!)
2. An element r ∈ R is called idempotent if r2 = r and nilpotent if rn = 0 for some n ∈ N.
(a) Find all units, zero divisors, idempotents and nilpotents in Z/3Z ⊕ Z/6Z.
(b) Is Z ⊕ Z an integral domain? More generally, if D is an integral domain, is D ⊕ D an integral
domain?
(c) Let S = {0, 2, 4, 6, 8}. Show that S with the operations of addition and multiplication modulo 10
is a field.
3. Enjoy reading break!
Problems
4. Recall we had seen that if we consider g(x) = x2 − 2 ∈ Q[x], then the ring
Q[x]/g(x)Q[x] ≡ {a + bx : a, b ∈ Q}
was a field with distinct equivalence classes for each q ∈ Q while squaring the equivalence
class √
of x
√
– the equivalence class of x2 – was the equivalence class of 2, which seems a lot like 2 and Q( 2).
Let’s investigate this some more.
(a) Let f (x) = x3 − 2. Show that f (x) is irreducible in Q[x]. Does f (x) have any roots in R? Find
them.
(b) Consider F := Q[x]/f (x)Q[x]. Show that the equivalence class of x ‘looks like’ a root in part (a),
i.e., what happens when you cube the equivalence class of x?
(c) Show that F is a vector space of dimension 3 over Q. (Note, all the vector space axioms hold
from the ring structure, you need only show that the dimension is 3.)
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(d) Show that the equivalence class of x2 does not correspond to a root of f (x), and F does not
contain all 3 (complex) roots of f (x). (Does this seem unreasonable given your answer in part
(a)?)
5. Let R be a ring. A subset S ⊂ R is a subring of R if S is itself a ring with the operations of R. In
particular, the operations descend to binary operations
+:S×S →S
×:S×S →S
on S, and all of the ring axioms are satisfied for elements of S with the operations.
(a) [Subring test] Show that a non-empty subset S ⊂ R is a subring of R if for all r, s ∈ S we have
r − s ∈ S and rs ∈ S. (This makes it easier to verify a set is a ring, if you know the set lives in a
larger ring.)
(b) Show that Z[i] := {a + bi : a, b ∈ Z}, the set of Gaussian integers, is an integral domain with the
usual addition and multiplication operations.
(c) Prove that the intersection of any collection of subrings of a ring R is a subring of R, but
√
Z[ 2] ∪ Z[i]
is not a subring of C.
Food for thought
We have now seen more (important) examples of rings, e.g.,
√
√
Q[ 2] = {a + b 2 : a, b ∈ Q},
√
√
Z[ 2] = {a + b 2 : a, b ∈ Z},
both of which can be viewed as the equivalence classes of polynomials in Q[x] and Z[x] respectively, modulo
x2 − 2.
√
But the integers
are very special rational numbers, and there is a way to think about Z[ 2] being the
√
‘integers’ of Q[ 2]. Can you see how? (See problem set 4, exercise 2(d).)
Practice problems
(1) Let R be a ring, and r ∈ R. If r4 = r2 , show that r2n = r2 for all n ∈ N. (By Shifrin’s convention that
all rings contain a multiplicative identity, all rings contain at least one element r such that r4 = r2 ,
but in some rings there are more such elements. Can you think of one?)
(2) Let S be a ring. Show that a2 − b2 = (a − b)(a + b) if and only if S is commutative.
(3) Let T be a ring, and suppose we have s, t ∈ T such that s a unit and t2 = 0. Show that s + t is also a
unit.
(4) Shifrin section 3.2: 2, 4, 10, 11, 16
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